Abstract
The general diophantine equations of the second and third degree are far from being totally solved. The equations considered in this paper are urn:x-wiley:01611712:media:ijmm317827:ijmm317827-math-0001 Infinitely many solutions of each of these equations will be stated explicitly, using the results from the ACF discussed before.It is known that the solutions of Pell′s equation are well exploited. We include it here because we shall use a common method to solve these three above mentioned equations and the method becomes very simple in Pell′s equations case.Some new third and fifth degree combinatorial identities are derived from units in algebraic number fields.
Highlights
THE CUBIC DIOPHANTINE EQUATIONSWe shall need formulas (0.6), (0.7), (0.8), (0.9) for n=3, viz. A2(n+3)=A0(n+l) +3DA0(n+2)+3D2A0(n+3)
Many solutions of each of these equations will be stated explicitly, using the results from the ACF discussed before
Some new third and fifth degree combinatorial identities are derived from units in algebraic number fields
Summary
We shall need formulas (0.6), (0.7), (0.8), (0.9) for n=3, viz. A2(n+3)=A0(n+l) +3DA0(n+2)+3D2A0(n+3). We shall need formulas (0.6), (0.7), (0.8), (0.9) for n=3, viz. Al(i), A(2i), Substituting in (3.1) the values for i n+3 from (3.2), we obtain, after simple rearrangements. We leave it to the reader to expand the determinant in (3.4) to obtain the Diophantine equation of the third degree as x3 + (9D3+I )y3 + z3 + (9D3_3)xyz + 6Dx2y + 3D2x2z. Even for D=l, equation (3.5) has a complicated form as x5 + lOy3 + z5 + 6xyz + 6x2y + 3x2z + 12y2x. For larger values of D and n the verification of (3.5) is only possible by computer, and without knowing (3.3) even a computer would have its problems. Diophantine equation which can be regarded as, and in a certain case represents, a generalization of Pell’s equation to the third degree
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